A Model of Porous Catalyst Accounting for Incipiently Non-isothermal Effects*

An approximate model accounting for incipiently non-isothermal effects is derived from a well-known model of porous catalyst for appropriate, realistic limiting values of the parameters. In this limit, the original model is a singularly perturbed, m-D reaction–diffusion system, and the approximate model is given by the m-D heat equation with nonlinear boundary condition, coupled with infinitely many (ifm2) 1-D semilinear parabolic equations, one for each point of the boundary of the spatial domain. Some limiting cases are still considered in the approximate model that lead to further simplifications

Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance

We consider the 2:1 internal resonances (such that Ω1>0 and Ω2 ≃ 2Ω1 are natural frequencies) that appear in a nearly inviscid, axisymmetric capillary bridge when the slenderness Λ is such that 0<Λ<π (to avoid the Rayleigh instability) and only the first eight capillary modes are considered. A normal form is derived that gives the slow evolution (in the viscous time scale) of the complex amplitudes of the eigenmodes associated with Ω1 and Ω2, and consists of two complex ODEs that are balances of terms accounting for inertia, damping, detuning from resonance, quadratic nonlinearity, and forcing. In order to obtain quantitatively good results, a two-term approximation is used for the damping rate. The coefficients of quadratic terms are seen to be nonzero if and only if the eigenmode associated with Ω2 is even. In that case the quadratic normal form possesses steady states (which correspond to mono- or bichromatic oscillations of the liquid bridge) and more complex periodic or chaotic attractors (corresponding to periodically or chaotically modulated oscillations). For illustration, several bifurcation diagrams are analyzed in some detail for an internal resonance that appears at Λ ≃ 2.23 and involves the fifth and eighth eigenmodes. If, instead, the eigenmode associated with Ω2 is odd, and only one of the eigenmodes associated with Ω1 and Ω2 is directly excited, then quadratic terms are absent in the normal form and the associated dynamics is seen to be fairly simple

Multi-scale path planning for a planetary exploration vehicle with multiple locomotion modes

Planetary exploration vehicles (rovers) can encounter with a great variety of situations. Most of them are related to the terrain, which can cause the end of the mission if these vehicles are not able to traverse it. It was the case of Spirit rover, which got stuck in loose sand, making it impossible to continue advancing. A solution to this is to make rovers capable of modifying their locomotion to traverse terrains with particular terramechanic parameters.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Faraday instability threshold in large-aspect-ratio containers

We consider the Floquet linear problem giving the threshold acceleration for the appearance of Faraday waves in large-aspect-ratio containers, without further restrictions on the values of the parameters. We classify all distinguished limits for varying values of the various parameters and simplify the exact problem in each limit. The resulting simplified problems either admit closed-form solutions or are solved numerically by the well-known method introduced by Kumar & Tuckerman (1994). Some comparisons are made with (a) the numerical solution of the original exact problem, (b) some ad hoc approximations in the literature, and (c) some experimental results

The Asymptotic Justification of a Nonlocal 1-D Model Arising in Porous Catalyst Theory

An asymptotic model of isothermal catalyst is obtained from a well-known model of porous catalyst for appropriate, realistic limiting values of some nondimensional parameters. In this limit, the original model is a singularly perturbedm-D reaction–diffusion system. The asymptotic model consists of an ordinary differential equation coupled with a semilinear parabolic equation on a semi-infinite one-dimensional interval

Viscous Faraday waves in two-dimensional large-aspect-ratio containers

A weakly nonlinear analysis of one-dimensional viscous Faraday waves in two-dimensional large-aspect-ratio containers is presented. The surface wave is coupled to a viscous long-wave mean flow that is slaved to the free-surface deformation. The relevant Ginzburg–Landau-like amplitude equations are derived from first principles, and can be of three different types, depending on the ratio between wavelength, depth and the viscous length. These three equations are new in the context of Faraday waves. The coefficients of these equations are calculated for arbitrary viscosity and compared with their counterparts in the literature for small viscosity; a discrepancy in the cubic coefficient is due to a dramatic sensitivity of this coefficient on a small wavenumber shift due to interplay between viscous effects and parametric forcing

Weakly Nonuniform Thermal Effects in a Porous Catalyst: Asymptotic Models and Local Nonlinear Stability of the Steady States.

This paper considers a first-order, irreversible exothermic reaction in a bounded porous catalyst, with smooth boundary, in one, two, and three space dimensions. It is assumed that the characteristic reaction time is sufficiently small for the chemical reaction to be confined to a thin layer near the boundary of the catalyst, and that the thermal diffusivity is large enough for the temperature to be uniform in the reaction layer, but that it is not so large as to avoid significant thermal gradients inside the catalyst. For appropriate realistic limiting values of the several nondimensional parameters of the problem, several time-dependent asymptotic models are derived that account for the chemical reaction at the boundary (that becomes essentially impervious to the reactant), heat conduction inside the catalyst, and exchange of heat and reactant with the surrounding unreacted fluid. These models possess asymmetrical steady states for symmetric shapes of the catalyst, and some of them exhibit a rich dynamic behavior that includes quasi-periodic phenomena. In one case, the linear stability of the steady states, and also the local bifurcation to quasi-periodic solutions via center manifold theory and normal form reduction, are analyzed

Standing wave description of nearly conservative, parametrically excited waves in extended systems

We consider the standing wavetrains that appear near threshold in a nearly conservative, parametrically excited, extended system that is invariant under space translations and reflection. Sufficiently close to threshold, the relevant equation is a Ginzburg-Landau equation whose cubic coefficient is extremely sensitive to wavenumber shifts, which can only be understood in the context of a more general quintic equation that also includes two cubic terms involving the spatial derivative. This latter equation is derived from the standard system of amplitude equations for counterpropagating waves, whose validity is well established today. The coefficients of the amplitude equation for standing waves are obtained for 1D Faraday waves in a deep container, to correct several gaps in former analyses in the literature. This application requires to also consider the effect of the viscous mean flow produced by the surface waves, which couples the dynamics of the surface waves themselves with the free surface deformation induced by the mean flow